= Solution
Since the kernel has integral one, The <Tonelli theorem> and the <triangle inequality> give
$$
\|f*g-f\|_1\leq\int g(s)\left(\int|f(t-s)-f(t)|dt\right)ds\leq\sup_{|h|\leq\delta}\|f(\cdot+h)-f\|_1.
$$
This proves \b[the stated <convolution> approximation estimate].
For the unlabelled compactness assertion let $M$ bound the <norms> of $S$, and fix $\eta>0$. Choose the kernel width $\delta$ smaller than the translation radius supplied for $\eta$, so that the same bound holds for all shifts of size at most this width. The preceding estimate gives $\|f*g-f\|_1<\eta$ uniformly. The smoothed family has uniform bound $M/\delta$ and uniform Lipschitz constant $M/\delta^2$. Choose $K$ with all original tails outside $[-K,K]$ below $\eta$, and put $L=K+\delta$. The <Tonelli theorem> gives
$$
\int_{|t|\geq L}|(f*g)(t)|dt\leq\int g(s)\int_{|u|\geq K}|f(u)|du\,ds<\eta.
$$
On $[-L,L]$ the smoothed family is relatively compact in the uniform <norm> by the <Arzelà-Ascoli theorem>. Take a finite uniform net with error below $\eta/(2L)$ and extend its centers by zero outside that interval. It is a finite $L^1$ net for the smoothed-and-truncated family with error $\eta$. Each original $f$ is within another $2\eta$ of its corresponding truncated smoothing. Thus \b[$S$ is <totally bounded> in $L^1$]. The displayed (a),(b) are the two hypotheses used together in this argument.
For the final family, <Young convolution inequality> gives $\|f_0*f\|_1\leq\|f_0\|_1$. Its translations satisfy
$$
\|\tau_h(f_0*f)-f_0*f\|_1\leq\|\tau_hf_0-f_0\|_1\|f\|_1\leq\|\tau_hf_0-f_0\|_1,
$$
which tends to zero uniformly over the family. Since $f$ vanishes outside $[-1,1]$,
$$
\int_{|t|\geq K}|(f_0*f)(t)|dt\leq\|f\|_1\int_{|u|\geq K-1}|f_0(u)|du\longrightarrow0
$$
uniformly as well. Both compactness hypotheses apply, proving \b[$S_0$ is <totally bounded>].
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